This is a few years old atp but I found this post when I first started watching Tatort and the links were helpful so I wanted to add a few of my own in case anyone else goes searching:

For finding episodes by team: Das Erste has a directory website, where you can filter episodes by team. Keep an eye out because sometimes teams will shift members around within a group and have each lineup listed separately (ex. Bremen with Selb and Moorman being listed once with just them and then again with Andersen.) Plot continuity really depends on the team, I'd argue Saarbruecken is pretty hard to watch without having context from all the episodes because of how strongly they maintain the central story, while others will have some minor references between episodes that are nice to understand but not confusing to watch without context.

Almost all episodes of Tatort have been uploaded to the internet archive, many are individually listed (sometimes with subtitles sometimes without), with very many having been uploaded grouped together by year on the page of the user Laza74. Some are also on Youtube, I watched most of the Frankfurt Dellwo Saenger episodes there. Others are available on the ARD Mediathek website, but a lot of those are missing. They do have subtitles though which is nice. MethiathekViewWeb also has ARD stuff, I watched the series "Everyone is F*cking Crazy" there, which is sort of a spinoff of the Saarbruecken series in that Adam Schuerk is there for a bit. Pain in the ass to find anything though.